In this paper we are exclusively interested in dcpos and the Scott topology on them. When we say “(closed) irreducible” in this context we always mean (closed) irreducible with respect to the Scott topology. It is a fact that every directed set in a dcpo is irreducible in this sense, while a set that is directed and closed is of course a principal ideal, that is, of the form \(\mathord {\downarrow }x\) . Peter Johnstone discovered in 1981, [1]}, that a dcpo may have irreducible sets which are not directed, and closed irreducible sets which are not principal ideals; we will discuss his famous example in Section  and someone not familiar with it may want to have a look at it before reading on.
